# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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### dirichlet problem for laplace's equation

How can we show that a Dirichlet problem for Laplace's equation in a finite region has a unique solution. Usually we can consider u2 - u1, a difference in values.
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### Kato inequality [closed]

For any real-valued smooth function $u$, we have the Kato inequality $|\nabla|\nabla u||^2\leq(\operatorname{trace}(\operatorname{Hess}(u)))^2$, which holds when $|\nabla u|\neq0$. If moreover $u$ ...
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### Mean-value formula for inhomogeneous harmonic functions

I am working on Evans' PDE textbook problems, but I am stuck with the following problem about modification of the proof of the mean-value formula for harmonic functions. I cannot really see how to ...
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### $|Du|^2$ is subharmonic if $u$ is harmonic.

In Evan's textbook "Partial Differential Equation", question 5 in section 2.5 says "$|Du|^2$ is subharmonic if $u$ is harmonic.". This can be easily proven, but do we really need the derivative $D$? I ...
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### I need to prove that this function is harmonic! [Solved]

I need to prove that $u:\mathbb{R}\times(-\frac{\pi}{2},\frac{\pi}{2})\rightarrow\mathbb{R}$ $$u(x,y)=\sum_{n \ \text{ is odd}}\cos(ny)e^{n(x-n)}$$ is harmonic. I have no idea which theorem or ...
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### Laplace's Equation in Polar Coordinates

I am trying to express Laplace's equation in terms of polar coordinates. That is, $$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0,\\ x=r\cos\theta,\\ y=r\sin\theta.$$ My book ...
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### Let $f$ be a harmonic function. Prove that $\overline{f}$ is harmonic.

Let $f$ be a harmonic function. Prove that $\overline{f}$ is harmonic. I need help to write a rigorous proof. Thank you
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